<?xml version="1.0" encoding="ISO-8859-1"?>

<!DOCTYPE proofs SYSTEM "proof3.dtd">

<?xml-stylesheet type="text/xsl" href="proof3.xsl"?>

<proofs number_of_proofs="1">

<source>identity2.out</source>

<heading><![CDATA[
Prover9 (32) version June-2006C, June 2006.
Process 13015 was started by mccune on cleo.thornwood,
Mon Jun 19 16:40:10 2006
The command was "/home/mccune/bin/prover9 -f identity2.in".
]]></heading>

<proof number="1" length="90" max_count="65">

<comments><![CDATA[
% Proof 1 at 8.84 (+ 0.03) seconds.
% Length of proof is 90.
% Level of proof is 16.
% Maximum clause weight is 65.
% Given clauses 121.
]]></comments>

  <clause id="10">
    <literal><![CDATA[
      (x * e) * x = x
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="11">
    <literal><![CDATA[
      x * (x * y) = y
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="12">
    <literal><![CDATA[
      (x * y) * (z * u) = (x * z) * (y * u)
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="13">
    <literal><![CDATA[
      ((x * x) * x) * x = e
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="16">
    <literal><![CDATA[
      (x / y) * y = x
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="17">
    <literal><![CDATA[
      (x * y) / y = x
    ]]></literal>
    <justification jstring="[input].">
      <j1 rule="input"/>
    </justification>
  </clause>

  <clause id="18">
    <literal><![CDATA[
      ((c4 * c2) * c3) * c1 != ((c1 * c2) * c3) * c4
    ]]></literal>
    <justification jstring="[clausify].">
      <j1 rule="clausify"/>
    </justification>
  </clause>

  <clause id="19">
    <literal><![CDATA[
      e * e = e
    ]]></literal>
    <justification jstring="[para(10(a,1),10(a,1,1)),demod(11(5)),flip(a)].">
      <j1 rule="para" parents="10 10"/>
      <j2 rule="demod" parents="11"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="20">
    <literal><![CDATA[
      ((x * y) * x) * (e * y) = x * y
    ]]></literal>
    <justification jstring="[para(12(a,1),10(a,1))].">
      <j1 rule="para" parents="12 10"/>
    </justification>
  </clause>

  <clause id="21">
    <literal><![CDATA[
      ((x * e) * y) * (x * z) = x * (y * z)
    ]]></literal>
    <justification jstring="[para(10(a,1),12(a,1,1)),flip(a)].">
      <j1 rule="para" parents="10 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="22">
    <literal><![CDATA[
      (x * (y * e)) * (z * y) = (x * z) * y
    ]]></literal>
    <justification jstring="[para(10(a,1),12(a,1,2)),flip(a)].">
      <j1 rule="para" parents="10 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="23">
    <literal><![CDATA[
      (x * y) * ((x * z) * (y * u)) = z * u
    ]]></literal>
    <justification jstring="[para(12(a,1),11(a,1,2))].">
      <j1 rule="para" parents="12 11"/>
    </justification>
  </clause>

  <clause id="24">
    <literal><![CDATA[
      (x * y) * ((x * z) * u) = z * (y * u)
    ]]></literal>
    <justification jstring="[para(11(a,1),12(a,1,1)),flip(a)].">
      <j1 rule="para" parents="11 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="25">
    <literal><![CDATA[
      (x * y) * (z * (y * u)) = (x * z) * u
    ]]></literal>
    <justification jstring="[para(11(a,1),12(a,1,2)),flip(a)].">
      <j1 rule="para" parents="11 12"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="28">
    <literal><![CDATA[
      ((x * x) * x) * e = x
    ]]></literal>
    <justification jstring="[para(13(a,1),11(a,1,2))].">
      <j1 rule="para" parents="13 11"/>
    </justification>
  </clause>

  <clause id="36">
    <literal><![CDATA[
      (x / y) * x = y
    ]]></literal>
    <justification jstring="[para(16(a,1),11(a,1,2))].">
      <j1 rule="para" parents="16 11"/>
    </justification>
  </clause>

  <clause id="40">
    <literal><![CDATA[
      x / (y * x) = y
    ]]></literal>
    <justification jstring="[para(11(a,1),17(a,1,1))].">
      <j1 rule="para" parents="11 17"/>
    </justification>
  </clause>

  <clause id="42">
    <literal><![CDATA[
      e / x = (x * x) * x
    ]]></literal>
    <justification jstring="[para(13(a,1),17(a,1,1))].">
      <j1 rule="para" parents="13 17"/>
    </justification>
  </clause>

  <clause id="46">
    <literal><![CDATA[
      (x * y) * (e * (y * x)) = x
    ]]></literal>
    <justification jstring="[para(11(a,1),20(a,1,1,1)),demod(11(7))].">
      <j1 rule="para" parents="11 20"/>
      <j2 rule="demod" parents="11"/>
    </justification>
  </clause>

  <clause id="52">
    <literal><![CDATA[
      (x * (x / y)) * (e * y) = x
    ]]></literal>
    <justification jstring="[para(16(a,1),20(a,1,1,1)),demod(16(7))].">
      <j1 rule="para" parents="16 20"/>
      <j2 rule="demod" parents="16"/>
    </justification>
  </clause>

  <clause id="54">
    <literal><![CDATA[
      x * ((x * x) * x) = (x * x) * x
    ]]></literal>
    <justification jstring="[para(36(a,1),10(a,1,1)),demod(42(2),42(5))].">
      <j1 rule="para" parents="36 10"/>
      <j2 rule="demod" parents="42 42"/>
    </justification>
  </clause>

  <clause id="60">
    <literal><![CDATA[
      x / e = (x * x) * x
    ]]></literal>
    <justification jstring="[para(13(a,1),40(a,1,2))].">
      <j1 rule="para" parents="13 40"/>
    </justification>
  </clause>

  <clause id="64">
    <literal><![CDATA[
      ((x * e) * y) * z = x * (y * (x * z))
    ]]></literal>
    <justification jstring="[para(11(a,1),21(a,1,2))].">
      <j1 rule="para" parents="11 21"/>
    </justification>
  </clause>

  <clause id="65">
    <literal><![CDATA[
      (x * (y * (x * z))) * ((x * u) * v) = (x * (y * u)) * (z * v)
    ]]></literal>
    <justification jstring="[para(21(a,1),12(a,1,1)),demod(64(8)),flip(a)].">
      <j1 rule="para" parents="21 12"/>
      <j2 rule="demod" parents="64"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="70">
    <literal><![CDATA[
      ((x * x) * x) * (y * x) = (x * y) * e
    ]]></literal>
    <justification jstring="[para(13(a,1),21(a,1,2)),demod(28(4)),flip(a)].">
      <j1 rule="para" parents="13 21"/>
      <j2 rule="demod" parents="28"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="72">
    <literal><![CDATA[
      ((x * x) * x) * (y * z) = (x * y) * (((x * x) * x) * z)
    ]]></literal>
    <justification jstring="[para(16(a,1),21(a,1,1,1)),demod(60(3),60(7)),flip(a)].">
      <j1 rule="para" parents="16 21"/>
      <j2 rule="demod" parents="60 60"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="84">
    <literal><![CDATA[
      x * ((x * e) * e) = x
    ]]></literal>
    <justification jstring="[para(42(a,1),40(a,1)),demod(64(8),11(5))].">
      <j1 rule="para" parents="42 40"/>
      <j2 rule="demod" parents="64 11"/>
    </justification>
  </clause>

  <clause id="85">
    <literal><![CDATA[
      (x * e) * e = x * x
    ]]></literal>
    <justification jstring="[para(84(a,1),11(a,1,2)),flip(a)].">
      <j1 rule="para" parents="84 11"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="91">
    <literal><![CDATA[
      (x * y) * x = e * (y * x)
    ]]></literal>
    <justification jstring="[para(11(a,1),22(a,1,1)),flip(a)].">
      <j1 rule="para" parents="11 22"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="101">
    <literal><![CDATA[
      (e * (x * x)) * x = e
    ]]></literal>
    <justification jstring="[para(22(a,2),13(a,1,1)),demod(11(3))].">
      <j1 rule="para" parents="22 13"/>
      <j2 rule="demod" parents="11"/>
    </justification>
  </clause>

  <clause id="112">
    <literal><![CDATA[
      ((c4 * c2) * c3) * c1 != ((c1 * c2) * (c4 * e)) * (c3 * c4)
    ]]></literal>
    <justification jstring="[para(22(a,2),18(a,2))].">
      <j1 rule="para" parents="22 18"/>
    </justification>
  </clause>

  <clause id="122">
    <literal><![CDATA[
      e * ((x * x) * e) = x
    ]]></literal>
    <justification jstring="[para(22(a,2),28(a,1,1)),demod(11(3),91(5))].">
      <j1 rule="para" parents="22 28"/>
      <j2 rule="demod" parents="11 91"/>
    </justification>
  </clause>

  <clause id="139">
    <literal><![CDATA[
      (e * (x * x)) * (y * z) = (x * y) * ((e * (x * x)) * z)
    ]]></literal>
    <justification jstring="[back_demod(72),demod(91(2),91(8))].">
      <j1 rule="back_demod" parents="72"/>
      <j2 rule="demod" parents="91 91"/>
    </justification>
  </clause>

  <clause id="141">
    <literal><![CDATA[
      (e * (x * x)) * (y * x) = (x * y) * e
    ]]></literal>
    <justification jstring="[back_demod(70),demod(91(2))].">
      <j1 rule="back_demod" parents="70"/>
      <j2 rule="demod" parents="91"/>
    </justification>
  </clause>

  <clause id="144">
    <literal><![CDATA[
      x * (e * (x * x)) = e * (x * x)
    ]]></literal>
    <justification jstring="[back_demod(54),demod(91(2),91(6))].">
      <j1 rule="back_demod" parents="54"/>
      <j2 rule="demod" parents="91 91"/>
    </justification>
  </clause>

  <clause id="168">
    <literal><![CDATA[
      (x * x) * y = x * ((y * e) * (x * (e * y)))
    ]]></literal>
    <justification jstring="[para(85(a,1),22(a,2,1)),demod(64(8)),flip(a)].">
      <j1 rule="para" parents="85 22"/>
      <j2 rule="demod" parents="64"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="191">
    <literal><![CDATA[
      ((x * y) * (x * y)) * y = e * (x * e)
    ]]></literal>
    <justification jstring="[para(101(a,1),23(a,1,2)),demod(91(4)),flip(a)].">
      <j1 rule="para" parents="101 23"/>
      <j2 rule="demod" parents="91"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="198">
    <literal><![CDATA[
      (x * x) * e = e * x
    ]]></literal>
    <justification jstring="[para(122(a,1),11(a,1,2)),flip(a)].">
      <j1 rule="para" parents="122 11"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="217">
    <literal><![CDATA[
      x * (((y / x) * z) * u) = z * (y * u)
    ]]></literal>
    <justification jstring="[para(36(a,1),24(a,1,1))].">
      <j1 rule="para" parents="36 24"/>
    </justification>
  </clause>

  <clause id="246">
    <literal><![CDATA[
      (x * x) * (y * x) = e * (y * e)
    ]]></literal>
    <justification jstring="[para(101(a,1),24(a,1,2)),demod(91(4)),flip(a)].">
      <j1 rule="para" parents="101 24"/>
      <j2 rule="demod" parents="91"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="261">
    <literal><![CDATA[
      ((c4 * c2) * c3) * c1 != (x * c3) * ((x * ((c1 * c2) * (c4 * e))) * c4)
    ]]></literal>
    <justification jstring="[para(24(a,2),112(a,2))].">
      <j1 rule="para" parents="24 112"/>
    </justification>
  </clause>

  <clause id="291">
    <literal><![CDATA[
      (x * e) * (x * e) = e * x
    ]]></literal>
    <justification jstring="[para(198(a,1),22(a,2)),demod(19(3))].">
      <j1 rule="para" parents="198 22"/>
      <j2 rule="demod" parents="19"/>
    </justification>
  </clause>

  <clause id="296">
    <literal><![CDATA[
      (x * y) * (e * x) = x * (y * e)
    ]]></literal>
    <justification jstring="[para(198(a,1),24(a,1,2))].">
      <j1 rule="para" parents="198 24"/>
    </justification>
  </clause>

  <clause id="297">
    <literal><![CDATA[
      (x * (y * y)) * ((x * z) * e) = z * (e * y)
    ]]></literal>
    <justification jstring="[para(198(a,1),24(a,2,2))].">
      <j1 rule="para" parents="198 24"/>
    </justification>
  </clause>

  <clause id="306">
    <literal><![CDATA[
      ((x / y) * z) * u = x * (z * (y * u))
    ]]></literal>
    <justification jstring="[para(16(a,1),25(a,1,1)),flip(a)].">
      <j1 rule="para" parents="16 25"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="312">
    <literal><![CDATA[
      (e * x) * y = e * (x * (e * y))
    ]]></literal>
    <justification jstring="[para(19(a,1),25(a,1,1)),flip(a)].">
      <j1 rule="para" parents="19 25"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="313">
    <literal><![CDATA[
      x * (y * (z * u)) = z * (y * (x * u))
    ]]></literal>
    <justification jstring="[para(36(a,1),25(a,1,1)),demod(306(6))].">
      <j1 rule="para" parents="36 25"/>
      <j2 rule="demod" parents="306"/>
    </justification>
  </clause>

  <clause id="315">
    <literal><![CDATA[
      x * (y * (z * (x * (y * u)))) = z * u
    ]]></literal>
    <justification jstring="[para(36(a,1),25(a,2,1)),demod(306(5))].">
      <j1 rule="para" parents="36 25"/>
      <j2 rule="demod" parents="306"/>
    </justification>
  </clause>

  <clause id="400">
    <literal><![CDATA[
      x * (y * (z * (x * u))) = z * (y * u)
    ]]></literal>
    <justification jstring="[back_demod(217),demod(306(3))].">
      <j1 rule="back_demod" parents="217"/>
      <j2 rule="demod" parents="306"/>
    </justification>
  </clause>

  <clause id="406">
    <literal><![CDATA[
      (x * y) * e = e * ((x * x) * (e * (y * x)))
    ]]></literal>
    <justification jstring="[back_demod(141),demod(312(5)),flip(a)].">
      <j1 rule="back_demod" parents="141"/>
      <j2 rule="demod" parents="312"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="408">
    <literal><![CDATA[
      (x * y) * (e * ((x * x) * (e * z))) = e * ((x * x) * (e * (y * z)))
    ]]></literal>
    <justification jstring="[back_demod(139),demod(312(5),312(12)),flip(a)].">
      <j1 rule="back_demod" parents="139"/>
      <j2 rule="demod" parents="312 312"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="461">
    <literal><![CDATA[
      ((x * y) * z) * u = x * (z * (e * ((y * x) * (e * u))))
    ]]></literal>
    <justification jstring="[para(46(a,1),25(a,1,1)),demod(312(4)),flip(a)].">
      <j1 rule="para" parents="46 25"/>
      <j2 rule="demod" parents="312"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="574">
    <literal><![CDATA[
      (x * c3) * ((x * ((c1 * c2) * (c4 * e))) * c4) != c4 * (c3 * (e * ((c2 * c4) * (e * c1))))
    ]]></literal>
    <justification jstring="[back_demod(261),demod(461(7)),flip(a)].">
      <j1 rule="back_demod" parents="261"/>
      <j2 rule="demod" parents="461"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="603">
    <literal><![CDATA[
      x * ((x * y) * (e * (y * (x * e)))) = e * (x * e)
    ]]></literal>
    <justification jstring="[back_demod(191),demod(461(4),296(6))].">
      <j1 rule="back_demod" parents="191"/>
      <j2 rule="demod" parents="461 296"/>
    </justification>
  </clause>

  <clause id="637">
    <literal><![CDATA[
      x * (y * (e * (z * (e * (x * (y * u)))))) = e * (z * (e * u))
    ]]></literal>
    <justification jstring="[para(52(a,1),23(a,1,2,1)),demod(461(6),36(3),312(12))].">
      <j1 rule="para" parents="52 23"/>
      <j2 rule="demod" parents="461 36 312"/>
    </justification>
  </clause>

  <clause id="862">
    <literal><![CDATA[
      (x * y) * (z * (y * x)) = e * (z * x)
    ]]></literal>
    <justification jstring="[para(91(a,1),25(a,2))].">
      <j1 rule="para" parents="91 25"/>
    </justification>
  </clause>

  <clause id="883">
    <literal><![CDATA[
      (x * x) * y = x * (e * (x * y))
    ]]></literal>
    <justification jstring="[back_demod(168),demod(862(8))].">
      <j1 rule="back_demod" parents="168"/>
      <j2 rule="demod" parents="862"/>
    </justification>
  </clause>

  <clause id="896">
    <literal><![CDATA[
      (x * y) * (e * (x * (e * (x * (e * z))))) = e * (x * (e * (x * (e * (y * z)))))
    ]]></literal>
    <justification jstring="[back_demod(408),demod(883(6),883(16))].">
      <j1 rule="back_demod" parents="408"/>
      <j2 rule="demod" parents="883 883"/>
    </justification>
  </clause>

  <clause id="897">
    <literal><![CDATA[
      (x * y) * e = e * (x * (e * (x * (e * (y * x)))))
    ]]></literal>
    <justification jstring="[back_demod(406),demod(883(9))].">
      <j1 rule="back_demod" parents="406"/>
      <j2 rule="demod" parents="883"/>
    </justification>
  </clause>

  <clause id="901">
    <literal><![CDATA[
      e * (x * e) = y * (e * (y * (x * y)))
    ]]></literal>
    <justification jstring="[back_demod(246),demod(883(3)),flip(a)].">
      <j1 rule="back_demod" parents="246"/>
      <j2 rule="demod" parents="883"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="922">
    <literal><![CDATA[
      e * (x * (e * (x * (e * (y * (e * (y * (z * x)))))))) = z * (e * y)
    ]]></literal>
    <justification jstring="[back_demod(297),demod(897(5),896(12),883(6))].">
      <j1 rule="back_demod" parents="297"/>
      <j2 rule="demod" parents="897 896 883"/>
    </justification>
  </clause>

  <clause id="1087">
    <literal><![CDATA[
      e * (x * (y * e)) = y * (x * e)
    ]]></literal>
    <justification jstring="[para(291(a,1),24(a,1,2)),demod(296(4)),flip(a)].">
      <j1 rule="para" parents="291 24"/>
      <j2 rule="demod" parents="296"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1095">
    <literal><![CDATA[
      (x * y) * (e * (y * (x * e))) = e * x
    ]]></literal>
    <justification jstring="[para(291(a,1),25(a,2))].">
      <j1 rule="para" parents="291 25"/>
    </justification>
  </clause>

  <clause id="1110">
    <literal><![CDATA[
      e * (x * e) = x * (e * x)
    ]]></literal>
    <justification jstring="[back_demod(603),demod(1095(7)),flip(a)].">
      <j1 rule="back_demod" parents="603"/>
      <j2 rule="demod" parents="1095"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="1247">
    <literal><![CDATA[
      (x * c3) * ((x * ((c1 * c2) * (c4 * e))) * c4) != c4 * (c3 * (e * ((c2 * e) * (c4 * c1))))
    ]]></literal>
    <justification jstring="[para(12(a,1),574(a,2,2,2,2))].">
      <j1 rule="para" parents="12 574"/>
    </justification>
  </clause>

  <clause id="1360">
    <literal><![CDATA[
      e * ((x * (e * y)) * z) = e * ((x * e) * (e * (y * z)))
    ]]></literal>
    <justification jstring="[para(19(a,1),65(a,1,2,1)),demod(312(8),11(8),312(12))].">
      <j1 rule="para" parents="19 65"/>
      <j2 rule="demod" parents="312 11 312"/>
    </justification>
  </clause>

  <clause id="1620">
    <literal><![CDATA[
      c1 * (c3 * (e * ((c4 * c1) * (e * ((c2 * e) * c4))))) != c4 * (c3 * (e * ((c2 * e) * (c4 * c1))))
    ]]></literal>
    <justification jstring="[para(23(a,1),1247(a,1,2,1)),demod(461(11))].">
      <j1 rule="para" parents="23 1247"/>
      <j2 rule="demod" parents="461"/>
    </justification>
  </clause>

  <clause id="1768">
    <literal><![CDATA[
      e * (x * (e * ((y * e) * (e * (y * (x * z)))))) = (y * e) * z
    ]]></literal>
    <justification jstring="[para(1110(a,1),23(a,1,2,1)),demod(312(8),1360(8))].">
      <j1 rule="para" parents="1110 23"/>
      <j2 rule="demod" parents="312 1360"/>
    </justification>
  </clause>

  <clause id="1771">
    <literal><![CDATA[
      (x * e) * (x * (e * (x * (e * (y * x))))) = y * (x * y)
    ]]></literal>
    <justification jstring="[para(1110(a,2),23(a,1,2)),demod(897(6),11(13))].">
      <j1 rule="para" parents="1110 23"/>
      <j2 rule="demod" parents="897 11"/>
    </justification>
  </clause>

  <clause id="2355">
    <literal><![CDATA[
      (x * y) * z = e * (x * (e * (x * (e * (y * (e * (x * (e * (x * (e * z))))))))))
    ]]></literal>
    <justification jstring="[para(144(a,1),25(a,1,1)),demod(312(7),883(8),312(13),883(14)),flip(a)].">
      <j1 rule="para" parents="144 25"/>
      <j2 rule="demod" parents="312 883 312 883"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2726">
    <literal><![CDATA[
      e * (x * (e * (x * (e * (x * (e * (x * (e * (x * (e * (x * (e * (y * x))))))))))))) = y * (x * y)
    ]]></literal>
    <justification jstring="[back_demod(1771),demod(2355(10),11(20))].">
      <j1 rule="back_demod" parents="1771"/>
      <j2 rule="demod" parents="2355 11"/>
    </justification>
  </clause>

  <clause id="2727">
    <literal><![CDATA[
      e * (x * (e * (x * (e * (x * (e * (x * (e * y)))))))) = e * (z * (x * (e * (x * (e * (x * (e * (z * y))))))))
    ]]></literal>
    <justification jstring="[back_demod(1768),demod(2355(9),11(14),11(11),11(13),11(15),2355(16),11(26)),flip(a)].">
      <j1 rule="back_demod" parents="1768"/>
      <j2 rule="demod" parents="2355 11 11 11 11 2355 11"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="2816">
    <literal><![CDATA[
      c1 * (c3 * (c4 * (e * (c4 * (e * (c1 * (e * (c4 * (e * (c4 * (e * (c2 * (e * (c2 * (e * (c2 * (e * (c2 * (e * c4))))))))))))))))))) != c4 * (c3 * (c2 * (e * (c2 * (e * (c2 * (e * (c2 * (e * (c4 * c1))))))))))
    ]]></literal>
    <justification jstring="[back_demod(1620),demod(2355(12),11(25),11(27),2355(24),11(43),2355(51),11(64),11(66))].">
      <j1 rule="back_demod" parents="1620"/>
      <j2 rule="demod" parents="2355 11 11 2355 11 2355 11 11"/>
    </justification>
  </clause>

  <clause id="3184">
    <literal><![CDATA[
      e * (x * (e * (x * (e * (x * (e * (x * (e * (x * (e * y)))))))))) = x * (e * (x * y))
    ]]></literal>
    <justification jstring="[back_demod(883),demod(2355(2))].">
      <j1 rule="back_demod" parents="883"/>
      <j2 rule="demod" parents="2355"/>
    </justification>
  </clause>

  <clause id="3531">
    <literal><![CDATA[
      x * (y * x) = y * (x * y)
    ]]></literal>
    <justification jstring="[back_demod(2726),demod(3184(19),11(7),11(6))].">
      <j1 rule="back_demod" parents="2726"/>
      <j2 rule="demod" parents="3184 11 11"/>
    </justification>
  </clause>

  <clause id="3548">
    <literal><![CDATA[
      x * (y * (x * y)) = y * x
    ]]></literal>
    <justification jstring="[para(3531(a,1),11(a,1,2))].">
      <j1 rule="para" parents="3531 11"/>
    </justification>
  </clause>

  <clause id="3570">
    <literal><![CDATA[
      x * (y * (z * y)) = z * (x * (y * x))
    ]]></literal>
    <justification jstring="[para(3531(a,1),313(a,1,2)),flip(a)].">
      <j1 rule="para" parents="3531 313"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="3595">
    <literal><![CDATA[
      x * (y * (z * (y * x))) = z * (x * y)
    ]]></literal>
    <justification jstring="[para(3548(a,1),313(a,1,2)),flip(a)].">
      <j1 rule="para" parents="3548 313"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="3615">
    <literal><![CDATA[
      x * (e * (y * (x * e))) = y * e
    ]]></literal>
    <justification jstring="[para(19(a,1),315(a,1,2,2,2,2))].">
      <j1 rule="para" parents="19 315"/>
    </justification>
  </clause>

  <clause id="3627">
    <literal><![CDATA[
      x * (y * (z * (u * (v * (x * (u * w)))))) = z * (y * (v * w))
    ]]></literal>
    <justification jstring="[para(315(a,1),313(a,1,2,2)),flip(a)].">
      <j1 rule="para" parents="315 313"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="3806">
    <literal><![CDATA[
      x * (y * (z * (x * (z * y)))) = y * z
    ]]></literal>
    <justification jstring="[para(400(a,2),3548(a,1))].">
      <j1 rule="para" parents="400 3548"/>
    </justification>
  </clause>

  <clause id="4648">
    <literal><![CDATA[
      x * (e * (y * (e * (y * (x * y))))) = e
    ]]></literal>
    <justification jstring="[para(901(a,1),3615(a,1,2,2)),demod(19(11))].">
      <j1 rule="para" parents="901 3615"/>
      <j2 rule="demod" parents="19"/>
    </justification>
  </clause>

  <clause id="4677">
    <literal><![CDATA[
      x * (y * (z * (e * (u * (e * (y * (x * v))))))) = z * (e * (u * (e * v)))
    ]]></literal>
    <justification jstring="[para(637(a,1),313(a,1,2)),flip(a)].">
      <j1 rule="para" parents="637 313"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="4957">
    <literal><![CDATA[
      e * (x * (e * (y * (e * (y * (e * (y * (e * (y * (e * x)))))))))) = x * (e * (y * y))
    ]]></literal>
    <justification jstring="[para(901(a,1),3570(a,1,2,2)),demod(2355(8),4648(14),19(8),11(9),3548(7),11(5),2355(8),11(18)),flip(a)].">
      <j1 rule="para" parents="901 3570"/>
      <j2 rule="demod" parents="2355 4648 19 11 3548 11 2355 11"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="5001">
    <literal><![CDATA[
      c1 * (c3 * (c4 * (e * (c4 * (e * (c1 * (c2 * c2))))))) != c4 * (c3 * (c2 * (e * (c2 * (e * (c2 * (e * (c2 * (e * (c4 * c1))))))))))
    ]]></literal>
    <justification jstring="[back_demod(2816),demod(4957(32),11(17),11(14))].">
      <j1 rule="back_demod" parents="2816"/>
      <j2 rule="demod" parents="4957 11 11"/>
    </justification>
  </clause>

  <clause id="5437">
    <literal><![CDATA[
      x * (e * (x * (e * (x * (e * (x * (e * y))))))) = e * (x * (e * (x * (e * (x * y)))))
    ]]></literal>
    <justification jstring="[para(1087(a,1),3595(a,1,2,2)),demod(2355(7),3806(13),11(9),11(8),2355(13),11(23),11(25)),flip(a)].">
      <j1 rule="para" parents="1087 3595"/>
      <j2 rule="demod" parents="2355 3806 11 11 2355 11 11"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="5609">
    <literal><![CDATA[
      c1 * (c3 * (c4 * (e * (c4 * (e * (c1 * (c2 * c2))))))) != c4 * (c3 * (e * (c2 * (e * (c2 * (e * (c2 * (c4 * c1))))))))
    ]]></literal>
    <justification jstring="[para(3595(a,2),5001(a,1,2,2,2,2,2,2)),demod(11(15),5437(38))].">
      <j1 rule="para" parents="3595 5001"/>
      <j2 rule="demod" parents="11 5437"/>
    </justification>
  </clause>

  <clause id="6028">
    <literal><![CDATA[
      e * (x * (y * (e * (y * (e * (y * (e * (x * z)))))))) = y * (e * (y * (e * (y * z))))
    ]]></literal>
    <justification jstring="[back_demod(2727),demod(5437(13),11(11)),flip(a)].">
      <j1 rule="back_demod" parents="2727"/>
      <j2 rule="demod" parents="5437 11"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="6706">
    <literal><![CDATA[
      c4 * (c3 * (e * (c2 * (e * (c2 * (e * (c2 * (c4 * c1)))))))) != c1 * (c3 * (c4 * (x * (e * (x * (c1 * (c2 * (e * (c2 * (e * (c2 * (c4 * x))))))))))))
    ]]></literal>
    <justification jstring="[para(922(a,2),5609(a,1,2,2,2,2)),demod(2355(21),2355(34),11(42),5437(47),11(38),11(40),11(42),2355(39),5437(50),4677(42),4677(36),11(31),2355(28),315(37),315(31),11(31),6028(29),11(24),11(25)),flip(a)].">
      <j1 rule="para" parents="922 5609"/>
      <j2 rule="demod" parents="2355 2355 11 5437 11 11 11 2355 5437 4677 4677 11 2355 315 315 11 6028 11 11"/>
      <j2 rule="flip"/>
    </justification>
  </clause>

  <clause id="6707">
    <literal><![CDATA[
      $F
    ]]></literal>
    <justification jstring="[resolve(6706,a,3627,a(flip))].">
      <j1 rule="resolve" parents="6706 3627"/>
    </justification>
  </clause>

</proof>

</proofs>
